The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single

The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single point by definition tells us something about the neighborhood of that point. Mentally expanding the series terms to their limit forms helps see that it's not just a random polynomial. I guess the real "miracle" is that for a large class of useful functions those ugly limits evaluate to simple forms.